Stimuli can be incorporated into dreams - for example, if someone in a sleep lab sees you are in REM sleep and sprays water on you, you're more likely to report having had a dream it was raining when you wake up. Yes, this has been formally tested. This provides strong evidence that dreams are going on during sleep.

More directly, communication has been established between dreaming and waking states by lucid dreamers in sleep labs. Lucid dreamers can make eye movements during their dreams to send predetermined messages to laboratory technicians monitoring them with EEGs. Again, this has been formally tested.